topos theory

# Contents

## Idea

A topos may be thought of as a generalized topological space. Accordingly, the notions of

have analogs for toposes and (∞,1)-toposes

## Definition

###### Definition

An object $A$ in a topos $ℰ$ is called a connected object if the hom-functor $ℰ\left(A,-\right)$ preserves finite coproducts.

Equivalently, an object $A$ is connected if it is nonempty (noninitial) and cannot be expressed as a coproduct of two nonempty subobjects.

###### Definition

A Grothendieck topos $ℰ$ is called a locally connected topos if every object $A\in ℰ$ is a coproduct of connected objects $\left\{{A}_{i}{\right\}}_{i\in I}$, $A={\coprod }_{i\in I}{A}_{i}$.

It follows that the index set $I$ is unique up to isomorphism, and we write

${\pi }_{0}\left(A\right)=I\phantom{\rule{thinmathspace}{0ex}}.$\pi_0(A) = I \,.

This construction defines a functor ${\Pi }_{0}:E\to \mathrm{Set}:A↦{\pi }_{0}\left(A\right)$ which is left adjoint to the constant sheaf functor, the left adjoint part of the global section geometric morphism.

Thus, for a locally connected topos we have

$\left({\Pi }_{0}⊣L\mathrm{Const}⊣\Gamma \right):ℰ\stackrel{\stackrel{{\Pi }_{0}}{\to }}{\stackrel{\stackrel{\mathrm{Const}}{←}}{\underset{\Gamma }{\to }}}\mathrm{Set}\phantom{\rule{thinmathspace}{0ex}}.$(\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{Const}{\leftarrow}}{\underset{\Gamma}{\to}}} Set \,.

This is the connected component functor. It generalises the functor, also denoted ${\pi }_{0}$ or ${\Pi }_{0}$, which to a topological space assigns the set of connected components of that space. See the examples below.

The following proposition asserts that the existence of ${\Pi }_{0}$ already characterizes locally connected toposes.

###### Proposition

A Grothendieck topos $ℰ$ is locally connected precisely if the global section geometric morphism $\Gamma :ℰ\to \mathrm{Set}$ is an essential geometric morphism $\left({\Pi }_{0}⊣L\mathrm{Const}⊣\Gamma \right):ℰ\to \mathrm{Set}$.

A proof appears as (Johnstone, lemma C.3.3.6).

###### Proof

Suppose that $\left({\Pi }_{0}⊣L\mathrm{Const}⊣\Gamma \right):ℰ\to \mathrm{Set}$ exists.

First notice that an object $A$ is connected in the above sense precisely if ${\Pi }_{0}\left(A\right)=*$.

Because for all $S\in \mathrm{Set}$ the connectivity condition demands that

$ℰ\left(A,\coprod _{S}L\mathrm{Const}*\right)\simeq \coprod _{S}ℰ\left(A,*\right)\simeq \coprod _{S}*\simeq S$\mathcal{E}(A, \coprod_S L Const *) \simeq \coprod_S \mathcal{E}(A,*) \simeq \coprod_S * \simeq S

but by the $\left({\Pi }_{0}⊣L\mathrm{Const}\right)$-hom-equivalence the first term is

$\cdots \simeq ℰ\left(A,L\mathrm{Const}\coprod _{S}*\right)\simeq \mathrm{Set}\left({\Pi }_{0}\left(A\right),S\right)$\cdots \simeq \mathcal{E}(A, L Const \coprod_S *) \simeq Set(\Pi_0(A), S)

and the last set is isomorphic to $S$ precisely for ${\Pi }_{0}\left(A\right)$ is the singleton set.

So we need to show that given the extra left adjoint ${\Pi }_{0}$, every object of $ℰ$ is a coproduct of objects for which ${\Pi }_{0}\left(-\right)$ is the point.

For that purpose consider for every object $A\in ℰ$ the pullback diagram

$\begin{array}{ccc}{i}_{A}^{*}{\underset{\to }{\mathrm{lim}}}_{{\Pi }_{0}\left(A\right)}*& \to & {\underset{\to }{\mathrm{lim}}}_{{\Pi }_{0}\left(A\right)}*\\ {}^{\simeq }↓& & {↓}^{\simeq }\\ A& \stackrel{{i}_{A}}{\to }& L\mathrm{Const}{\Pi }_{0}\left(A\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ i_A^* {\lim_\to}_{\Pi_0(A)} * &\to& {\lim_\to}_{\Pi_0(A)} * \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ A &\stackrel{i_A}{\to}& L Const \Pi_0 (A) } \,,

where the bottom morphism is the $\left(\Pi ⊣L\mathrm{Const}\right)$-unit and the right isomorphism is the identification of any set as the colimit (here: coproduct) of the functor over the set itself that is constant on the point. Since pullbacks of isomorphism are isomorphisms, also the left morphism is an iso.

By universal colimits this left morphism is equivalently

${\underset{\to }{\mathrm{lim}}}_{s\in {\Pi }_{0}\left(A\right)}\left({i}_{A}^{*}{*}_{s}\right)\stackrel{\simeq }{\to }A${\lim_\to}_{s \in \Pi_0(A)} (i_A^* *_s) \stackrel{\simeq}{\to} A

and hence expresses $A$ as a coproduct of objects ${i}_{A}*{*}_{s}$, each of which is a pullback

$\begin{array}{ccc}{i}_{A}^{*}{*}_{s}& \to & L\mathrm{Const}*\\ ↓& & {↓}^{s}\\ A& \stackrel{{i}_{A}}{\to }& L\mathrm{Const}{\Pi }_{0}A\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ i_A^* *_s &\to& L Const * \\ \downarrow && \downarrow^{\mathrlap{s}} \\ A &\stackrel{i_A}{\to}& L Const \Pi_0 A } \,,

where the right morphism includes the element $s$ into the set ${\Pi }_{0}A$. By applying ${\Pi }_{0}$ to this diagram and pasting on the $\left({\Pi }_{0}⊣L\mathrm{Const}\right)$-counit we get

$\begin{array}{ccccc}{\Pi }_{0}\left({i}_{A}^{*}{*}_{s}\right)& \to & {\Pi }_{0}L\mathrm{Const}*& \to & *\\ ↓& & {↓}^{}& & ↓\\ {\Pi }_{0}\left(A\right)& \stackrel{{\Pi }_{0}\left({i}_{A}\right)}{\to }& {\Pi }_{0}L\mathrm{Const}{\Pi }_{0}A& \to & {\Pi }_{0}A\end{array}$\array{ \Pi_0(i_A^* *_s) &\to& \Pi_0 L Const * &\to& * \\ \downarrow && \downarrow^{} && \downarrow \\ \Pi_0(A) &\stackrel{\Pi_0(i_A)}{\to}& \Pi_0 L Const \Pi_0 A &\to& \Pi_0 A }

and by the zig-zag identity the bottom morphism is the identity. This says that in

${\Pi }_{0}\left({\underset{\to }{\mathrm{lim}}}_{{\Pi }_{0}A}{i}_{A}^{*}{*}_{s}\stackrel{\simeq }{\to }A\right)\simeq \left({\underset{\to }{\mathrm{lim}}}_{{\Pi }_{0}A}{\Pi }_{0}\left({i}_{A}^{*}{*}_{s}\right)\stackrel{\simeq }{\to }{\Pi }_{0}\left(A\right)\right)$\Pi_0( {\lim_{\to}}_{\Pi_0 A} i_A^* *_s \stackrel{\simeq }{\to} A) \simeq ({\lim_\to}_{\Pi_0 A} \Pi_0(i_A^* *_s) \stackrel{\simeq}{\to} \Pi_0(A))

all the component maps out of the coproduct factor through the point. This means that this can only be an isomorphism if all these component maps are point inclusions, hence if ${\Pi }_{0}\left({i}_{A}^{*}{*}_{s}\right)\simeq *$ for all $s\in {\Pi }_{0}A$.

However, this doesn’t mean that essential geometric morphisms are the “relative” analog of locally connected toposes; in general one needs to impose an additional condition, which is automatic in the case of the global sections morphism, to obtain the notion of a locally connected geometric morphism.

## Properties

### Characterization over locally connected sites

See at locally connected site.

### Equivalent conditions

###### Definition

For $C$ and $C$ cartesian closed categories, a functor $F:C\to D$ that preserves products is called a cartesian closed functor if the canonical natural transformation

$F\left({B}^{A}\right)\to \left(F\left(B\right){\right)}^{F\left(A\right)}$F(B^A) \to (F(B))^{F(A)}

(which is the adjunct of $F\left(A\right)×F\left({B}^{A}\right)\simeq F\left(A×{B}^{A}\right)\to F\left(B\right)$) is an isomorphism.

###### Proposition

The constant sheaf-functor $\Delta :𝒮\to ℰ$ is a cartesian closed functor precisely if $ℰ$ is a locally connected topos.

### Locally connected and connected

A topos $E$ is called a connected topos if the left adjoint $L\mathrm{Const}:\mathrm{Set}\to E$ is a full and faithful functor.

###### Proposition

If $\Gamma :E\to \mathrm{Set}$ is a locally connected topos, then it is also a connected topos — in that $L\mathrm{Const}$ is full and faithful — if and only if the left adjoint ${\Pi }_{0}$ of $L\mathrm{Const}$ preserves the terminal object.

This is (Johnstone, C3.3.3).

Notice that for a connected and locally connected topos, the adjunction

$\mathrm{Set}\stackrel{\stackrel{{\Pi }_{0}}{←}}{↪}E$Set \stackrel{\overset{\Pi_0}{\leftarrow}}{\hookrightarrow} E

exhibits Set as a reflective subcategory of $E$. We may think then of Set as being the localization of $E$ at those morphisms that induce isomorphisms of connected components.

## Examples

###### Example

For $X$ a topological space, the category of sheaves $\mathrm{Sh}\left(X\right)≔\mathrm{Sh}\left(\mathrm{Op}\left(X\right)\right)$ is a locally connected topos precisely if $X$ is a locally connected space. The functor ${\Pi }_{0}$ sends a sheaf $F\in \mathrm{Sh}\left(X\right)$ to the set of connected components of the corresponding etale space.

###### Example

For $C=$ CartSp the site of Cartesian spaces with its good open cover coverage, the topos $\mathrm{Sh}\left(\mathrm{CartSp}\right)$ of smooth spaces is locally connected. An arbitrary $X\in \mathrm{Sh}\left(\mathrm{CartSp}\right)$ is sent to the colimit ${\mathrm{lim}}_{\to }X\in \mathrm{Set}$. If $X$ is a diffeological space or even a smooth manifold, then this is the set of connected components of the underlying topological space.

###### Example

Suppose that $C$ is a site such that constant presheaves on $C$ are sheaves. Then the left adjoint ${\Pi }_{0}$ exists and is given by the colimit functor: if we write $L:\mathrm{PSh}\left(C\right)\to \mathrm{Sh}\left(C\right)$ for sheafification, then for any sheaf $X$, we have

${\mathrm{Hom}}_{\mathrm{Sh}\left(C\right)}\left(X,L\mathrm{Const}S\right)\simeq {\mathrm{Hom}}_{\mathrm{PSh}\left(C\right)}\left(X,L\mathrm{Const}S\right)\simeq {\mathrm{Hom}}_{\mathrm{PSh}\left(C\right)}\left(X,\mathrm{Const}S\right)\simeq {\mathrm{Hom}}_{\mathrm{Set}}\left(\underset{\to }{\mathrm{lim}}X,S\right)\phantom{\rule{thinmathspace}{0ex}}.$Hom_{Sh(C)}(X, L Const S) \simeq Hom_{PSh(C)}(X, L Const S) \simeq Hom_{PSh(C)}(X, Const S) \simeq Hom_{Set}(\lim_\to X, S) \,.

In particular, this is the case if every covering sieve in $C$ is connected, i.e. $C$ is a locally connected site.

If $C$ furthermore has a terminal object $1$, then the global sections functor $\Gamma :\mathrm{Sh}\left(C\right)\to \mathrm{Set}$ (the right adjoint of $L\mathrm{Const}$) is simply given by evaluation at $1$, and so the unit $S\to \Gamma L\mathrm{Const}S\cong L\mathrm{Const}S\left(1\right)$ is an isomorphism. Thus in this case $\mathrm{Sh}\left(C\right)$ is additionally connected. This situation also applies to $C=\mathrm{CartSp}$ in example 2 above.

###### Example

If $C$ is a category with all finite limits and if the unique functor $\pi :C\to *$ to the terminal category preserves covers (for $*$ equipped with the trivial topology/coverage) then $\mathrm{Sh}\left(C\right)$ is locally connected. This is because the inclusion of the terminal object $i:*\to C$ provides a right adjoint to $\pi$, so that there is an adjoint quadruple of functors on presheaf categories

$\left({\pi }_{!}\simeq {\mathrm{Lan}}_{\pi }\right)⊣\left({\pi }^{*}\simeq {i}_{!}\simeq {\mathrm{Lan}}_{i}\right)⊣\left({\mathrm{pi}}_{*}\simeq {i}^{*}\right)⊣\left({\pi }^{!}\simeq {i}_{*}\simeq {\mathrm{Ran}}_{i}\right)\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\mathrm{PSh}\left(C\right)↔\mathrm{PSh}\left(*\right)\simeq \mathrm{Sh}\left(C\right)\simeq \mathrm{Set}\phantom{\rule{thinmathspace}{0ex}},$(\pi_! \simeq Lan_\pi) \dashv (\pi^\ast \simeq i_! \simeq Lan_i) \dashv (pi_\ast \simeq i^\ast ) \dashv (\pi^! \simeq i_* \simeq Ran_i) \;\colon\; PSh(C) \leftrightarrow PSh(\ast) \simeq Sh(C) \simeq Set \,,

where ${\mathrm{Lan}}_{\left(-\right)}$ and ${\mathrm{Ran}}_{\left(-\right)}$ denote let and right Kan extension, respectively. Now if $C\to *$ indeed preserves covers and using that $C\to *$ trivially preserves finite limits and hence is a flat functor, then by the discussion at morphism of sites the first three functors here descend to sheaves and hence exhibit $\mathrm{Sh}\left(C\right)$ as being locally connected.

But beware that the assumptions here are stronger than they may seem: that $C\to *$ preserves covers is not automatic, but is a strong condition. It is violated as soon as $C$ contains an empty object with empty cover, such as is the case in most categories of spaces, notably in categories of open subsets $\mathrm{Op}\left(X\right)$ of a topological space $X$, as in example 1.

## References

Section C1.5 and C3.3 of

A variant is in

Discussion of characterizations of sites of definition of locally connected toposes is in

• Olivia Caramello, Site characterizations for geometric invariants of toposes, Theory and Applications of Categories, Vol. 26, 2012, No. 25, pp 710-728. (TAC)

Revised on June 10, 2013 11:08:17 by Urs Schreiber (89.204.154.99)